Enzyme kinetics

Enzyme kinetics models how free enzyme, substrate, enzyme-substrate complexes, and product concentrations evolve through a reaction mechanism.

When each chemical reaction only changes the state of one enzyme molecule, summing all enzyme-state concentrations gives a conserved total. For sequential complexes satisfying ${\dot{C}}_0=-R_1$, ${\dot{C}}_i=R_i-R_{i+1}$, ${\dot{C}}_n=R_n$, the sum telescopes to zero. Bound substrate must be included separately in substrate mass conservation.

Measurement of early reaction behaviour before a steady-state distribution of enzyme intermediates is established. Bursts and lags can separate fast chemical conversion from slower steps.

An initial rapid accumulation of product followed by slower turnover. It can reveal a fast first conversion followed by a slower recycling step; converting burst amplitude to active-enzyme concentration requires the appropriate mechanistic assumptions.

Mixing reactants, allowing a controlled short reaction interval and then quenching chemistry for product measurement. It can help resolve chemical steps independently of later product release.

Rapid mixing followed by time-resolved observation, commonly by fluorescence or absorbance. A signal change must be assigned through a kinetic model rather than automatically equated with chemistry.

For simple Michaelis–Menten kinetics at low substrate concentration, $v/[E]\simeq (k_{\mathrm{cat}}/K_M)[S]$. This efficiency combines productive substrate capture and turnover; it should be interpreted with the assay conditions and mechanism.

Catalytic cycles per unit time per active enzyme site under defined saturating conditions. In the corresponding simple model, $k_{\mathrm{cat}}=V_{\max }/[E{]}_{\mathrm{active}}$.

A time course with rapid initial formation of a measured reaction product, followed by slower steady-state turnover. It can reveal fast initial chemistry followed by slow regeneration of competent enzyme; the monitored product and mixing time resolution are essential to interpretation.

An enzyme may bind one molecule of a substrate to form $C_1$, then a second to form $C_2$, with product-release reactions $C_1\to E+P$ and $C_2\to C_1+P$. The law of mass action gives conservation laws $e+c_1+c_2=e_0$ and $s+c_1+2c_2+p=s_0$. Scaling substrate by $s_0$, complexes by $e_0$ and time by $(k_1{e}_0{)}^{-1}$ exposes the small ratio $ϵ=e_0/s_0$: the substrate evolves on the slow scale while the complex equations have the form $ϵv_i^{\prime }=g_i$. This is a starting point for quasi-steady-state approximation when $ϵ\ll 1$.

Michaelis-Menten kinetics models substrate binding to an enzyme, complex dissociation and product formation. Under the relevant quasi-steady-state regime, its initial-rate law is the saturating relation given by the following equation.

Substrate replication competes with saturating enzyme clearance in this fast-slow model, where $K>\lambda >0$ and $ϵ\ll 1$. The physical strip $u\ge 0$, $0\le v\le 1$ is invariant. If $0<\eta <\lambda /K$, the origin is attracting and a positive saddle exists at $u_e=\lambda /\eta -K$, $v_e=1-\eta K/\lambda$. Eliminating the fast complex fraction gives $u^{\prime }\simeq \eta u(u-u_e)/(u+K)$, separating extinction from runaway. For initially zero complexes, the stable-manifold threshold is $u_{\ast }=u_e+ϵ(1-\eta )v_e+O({ϵ}^2)$, away from singular parameter limits. To derive the correction, write the stable manifold as $u=u_e+ϵ\varphi (v)$; its invariance equation gives ${\varphi }^{\prime }=-(1-\eta )$ and $\varphi (v_e)=0$. A Dulac factor $1/u$ has strictly negative weighted divergence, excluding a periodic alternative to the two basins. Saturated clearance and autocatalytic growth can therefore create an unstable concentration threshold rather than a stable nonzero balance.

A plot of $[S]/v$ against $[S]$ for a reaction modeled by Michaelis-Menten kinetics. The slope is $1/V_{\max }$, the vertical intercept $K_m/V_{\max }$ and the horizontal intercept $-K_m$; graphical transformation changes how measurement errors are weighted.

The Michaelis-Menten rate law isIt is linear at low substrate concentration and saturates at $V_{\max }$ at high concentration.

The substrate concentration giving half the limiting reaction rate in the Michaelis-Menten equation. In the elementary mechanism $E+S\rightleftharpoons ES\to E+P$, it is $K_M=(k_{-1}+k_{\mathrm{cat}})/k_1$ under the quasi-steady-state approximation, whereas the binding dissociation constant is $K_d=k_{-1}/k_1$.

When enzyme complexes relax much faster than substrate concentrations, the quasi-steady-state approximation sets the time derivatives of the fast complexes to zero and solves their algebraic balance equations.

It approximates the fast intermediate by a steady state balance without requiring the entire chemical system to be at equilibrium.

A quasi-steady-state approximation for enzyme-substrate complexes can remain useful when total enzyme and initial substrate concentrations are comparable. The relevant parameter can involve the sum $s_0+K_m$, rather than $s_0$ alone. In a mechanism-based inhibition model with $s_0/K_m=O(\epsilon )$, bounded intermediate decay rates and an inactivation rate of order $\epsilon$, the fast complexes adjust in a layer of scaled width $O(\epsilon )$. On bounded slow-time intervals the main complex follows the substrate, the later complex and accumulated inactivated enzyme remain $O(\epsilon )$, and the substrate has leading exponential depletion. An outer algebraic complex concentration does not satisfy the unbound initial condition inside that initial layer.

For sequential binding with dimensionless parameters $\alpha ,\beta ,\gamma$, the reduced product rate has the formIt is quadratic at low substrate concentration and saturates at high concentration.